Acoustic reflection coefficient logging

ABSTRACT

An illustrative embodiment of the present invention includes method and apparatus for performing acoustic reflection coefficient logging in a well borehole. An acoustic well logging tool having at least a pair of biradially mounted receivers disposed in a zero spacing configuration and each comprising a plurality of arcuate segments disposed in a one-to-one relationship with each other is moved through the well bore. The electrical outputs of corresponding receiver segment pairs are processed to extract their relative amplitude and phase at a plurality of frequencies over the acoustic spectrum. The amplitude and phase data are used to compute the acoustic reflection coefficient exactly at the said plurality of frequencies over the acoustic spectrum thereby providing a frequency spectrum of acoustic reflection coefficients as measured at each of the paired biradially mounted receiver segments at each borehole depth. Such acoustic reflection coefficient spectra may be utilized, among other things, for evaluating cement conditions in the well bore.

United States Patent Ingram [54] ACOUSTIC REFLECTION COEFFICIENT LOGGING[72] Inventor: John D. Ingram, Houston, Tex.

[73] Assignee: Schlumberger Technology Corporation, New York, NY.

[22] Filed: July 9, 1969 [21] Appl. No.: 840,179

[52] US. Cl ..340/15.5 A, 181/.5 AC [51] Int. Cl. ..G0lv 1/22 [58] Fieldof Search ..l8l/.5;340/15.5

[56] References Cited UNITED STATES PATENTS 3,136,381 6/1964 Anderson.,l8l/.5 3,215,934 11/1965 Sallen ..324/77 3,292,143 12/1966 Russell..340/15.5 3,308,426 3/1967 Wilson ..340/l8 3,311,876 3/1967 Lee..340/18 3,454,924 7/1969 Sherwood et al. ..340/l5.5

FOREIGN PATENTS OR APPLICATIONS 793,617 7/1966 Canada ..340/1 3 755,2733/1967 Canada ..340/9 [451 Oct. 10,1972

Primary Examiner-Benjamin A. Borchelt Assistant Examiner-N. MoskowitzAttorney-Ernest R. Archambeau, Jr., Stewart F. Moore, David L. Moseley,Edward M. Roney and William R. Sherman s7 7 ABSTRACT An illustrativeembodiment of the present invention includes method and apparatus forperforming acoustic reflection coefficient logging in a well borehole.An acoustic well logging tool having at least a pair of biradiallymounted receivers disposed in a zero spacing configuration and eachcomprising a plurality of arcuate segments disposed in a one-to-onerelationship with each other is moved through the well bore. Theelectrical outputs of corresponding receiver segment pairs are processedto extract their relative amplitude and phase at a plurality offrequencies over the acoustic spectrum. The amplitude and phase data areused to compute the acoustic reflection coefficient exactly at the saidplurality of frequencies over the acoustic spectrum thereby providing afrequency spectrum of acoustic reflection coefficients as measured ateach of the paired biradially mounted receiver seg ments at eachborehole depth. Such acoustic reflection coefficient spectra may beutilized, among other things, for evaluating cement conditions in thewell bore.

25 Claims, 13 Drawing Figures CONTROL CIRCUITS 92 I 93 I SIGNA l L /62CONTROL "PROCESSING -RE CIRCUITS CIRCUITS PATENTEDUBI 10 I972 SHEET 1[1F 6 F ORMA T/ON FIGI FREQ.

BORE- HOLE DEPTH FREQ.

FREQ.

John 0. Ingram FIG. 73

INVENT OR A TTORNEY PATENTEDIIBT 1 1912 3.697; 937

sum 2 or 6 REFLECT/ON COEFFICIENT F I G. 3 FREQ. (TOGO/sec) /o is 2'02'5 3b 3 5 40 4 5 5b 55 .60 REFLECT/ON COEFFICIENT CEMENTED UNCEMENTED II ANNULUS 5 fa 2 5 3b 422 4 5 522 5 5 FIG-4 FREQ (Kc) John D. Ingram INVE N TOR BY al a 3M4 ATTORNEY PATENTEI] URI 1 I972 SHEET 3 [IF 6CEMENTED UNCEMENTED ANNULUS REFLECTION COEFFICIENT FIGS I0 75 2O 3OFREQ. (KC) T'IRTATG/NG" DELAY RIC RIB

SIGNALS ON CONDUCTOR TIME FIG.

John D. Ingram INVENTOR A TTORNE Y PATENTEDucI 10 m2 SHEET 0F 6 FIG. 6

CONTROL CIRCUITS 0 b C dFeFfF F John D INGRA M INVENTOR BYLJ )3 ATTORNEYACOUSTIC REFLECTION COEFFICIENT LOGGING In a well completion, the stringof casing or pipe is set in a well bore and cement is forced into theannulus between the casing and the well bore primarily to separate oiland gas producing horizons from each other and from water bearingstrata. Obviously, if the cementing fails to provide a separation of onezone from another, then fluids under pressure from one zone may be ableto migrate and contaminate an otherwise productive nearby zone.Migration of water in particular produces undesirable water cutting of aproducing zone and possibly can make a well con-commercial.

It is difficult to obtain an accurate picture of conditions behind acasing because of the difficulty of propagating signals through themetal casing wall. Various prior proposals to determine a separationeffectiveness (i.e., the blocking or sealing characteristics) of thecement behind the casing have not been entirely successful indetermining the effective presence of cement in the annulus between thecasing and the formation. Further, it has not been possible to measurereliably the quality of the cement bond or seal between the casing andthe cement and the quality of the cement bond or seal between the cementand the formation using the methods of the various prior proposals.

The mere presence or absence of cement in the annulus between the casingand the formation is valuable, but incomplete information. While cementmay be present in the annulus, channels or inadequate sealing may stillpermit fluid communication between adjacent formations.

Use of the term bond in connection with the relationship of cement tothe casing or the formation is somewhat vague, since adherence along theentire boundary between the casing and the cement or between the cementand the formation is not necessary to prevent fluid communicationbetween adjacent porous zones. All that is necessary of a bond is thatthe relationship prevents the migration of fluids. Hereafter, referenceto bond" will mean that the separation of zones by cement is adequate toprevent fluid migration between the zones.

Several prior developments for obtaining the measure of the quality of acement bond relative to the casing have been disclosed in U.S. Pat. Nos.3,291,247; 3,291,248; and 3,292,246. These patents are all assigned tothe assignee of the present invention. These systems have generallyutilized acoustic principles where an acoustic signal is transmittedbetween a transmitter and a receiver. The amplitude of the early arrivalsignal (this early arrival is the casing signal since the acousticimpulse generally travels faster in the casing than in the surroundingcement of formation) at the receiver is measured as a determination ofthe quality of the adherence of cement to the casing. If a good contactexisted between the cement and easing the casing signal would beexpected to be attenuated because of the energy dissipated from thecasing to the cement and the surrounding formations. Whereas, if nocontact, or a poor bond existed the casing signal would be relativelyunattenuated. This procedure is sound enough, if a good cementcasingcontact exists but where a small space or annulus (sometimes called amicro-annulus) exists between the casing and the cement, such ameasurement can give an indication of a poor cement job when the cementis actually adequate. By adequate, it is meant that the micro-annuluseven though present does not permit fluid communication between adjacentporous formations. Moreover, such false indications of poor bonding caneasily be the case because of the manner in which a cement job isperformed.

In a primary cementing operation, cement is forced up the annulus aboutthe casing by relatively high pressure applied inside the casing. Thispressure tends to expand the casing. The pressure is maintained in thecasing while the cement is setting. Once the cement is set, the pressureis then released. Upon release of the pressure the casing can contract,thus forming a microannulus between the set cement and the casing.

A more refined technique for determining the quality of cement in theannulus between the casing and the formations is disclosed in U.S. Pat.No. 3,401,773 entitled, Method and Apparatus for Cement Logging of CasedBoreholes by Judson D. Synnott,lll which is assigned to the assignee ofthe present invention. In this technique, the amplitude of areverberated early (casing) signal arrival is recorded and additionally,the total energy of a selected later portion of the sonic signal isobtained by integration to provide a second indication of the quality orintegrity of the cement column. Even in the absence of a weak casingarrival, the additional step of observing the total energy obtained byintegrating a later portion of the signal in this manner can confirm thepresence of cement in the casing-formation annulus. Details of relatedmethods may also be had by reference to U.S. Pat. No. 3,401,772entitled, Method for Logging Cased Boreholes by Frank P. Kokesh, whichis assigned to the assignee of the present invention.

While the foregoing methods and apparatus provide very usefulinformation, it is desirable to more precisely determine the quality ofthe cement bond. For example, the energy content of the acoustic loggingsignals arriving at the receiver may depend on other factors than thecement bond to the casing or the integrity of the cement column,(sometimes called cement quality). Factors which can influence acousticenergy are: the formation hardness; eccentering of the acoustic loggingtool; construction materials of which the acoustic logging sonde ismade; type of casing; and the diameter of the borehole and the casing aswell as their shape or geometry.

The recognition of these difficulties and some anomalous results in thevarious prior proposals has led to the development of the method andapparatus of the present invention which can provide a log of thereflection coefficients which may be used to determine the cementeffectiveness under a range of well conditions. This result has beenaccomplished by the use of a quantative theoretical model of the cementproblem coupled with analysis of the theoretical results and comparisonof these results with acoustic logs taken from wells in the field.

Accordingly, it is an object of the present invention to provide a novelmethod and apparatus for determining exactly the acoustic reflectioncoefficient spectrum which may be used for determining cementingeffectiveness in a cased borehole.

It is a further object of the present invention to provide a method andapparatus for cement evaluation in a cased borehole which derives thecementing effectiveness by the evaluation of the reflection coefficientassociated with the radial propagation of acoustic energy is thesonde-well-bore-casing-annulus-formation system.

Another object of the present invention is to provide a method andapparatus for directly and exactly measuring the acoustic reflectioncoefficient in the mud column inside of a cased borehole.

A yet further object of the present invention is to provide a novelacoustic logging apparatus which may be utilized to directly measure therelative amplitude and phase of acoustic waves at two different radiiwithin a cased borehole and to utilize this data to compute the acousticreflection coefficient in the mud inside the casing in order todetermine the cementing effectiveness.

Briefly, in accordance with the objects of the present invention,methods and apparatus for evaluating the cement conditions in casedboreholes are provided. Novel acoustic logging apparatus is providedincluding an acoustic transmitter together with a plurality ofassociated segmented acoustic receiver pairs. The receiver pairs areused in a zero spacing configuration and are mounted in a biradialmanner. The biradial mounting and zero spacing configuration permitsamplitude and relative phase measurements to be made on reflected radialacoustic energy emanating from the acoustic transmitter and reflectedfrom material discontinuties in the system. These measurements may thenbe used to compute the acoustic reflection coefficient spectrum of thesonde-well-bore-casing-annulusformation system directly. A novelcomputer apparatus for performing the computations necessary to extractthe acoustic reflection coefficients from the measurements is disclosedalong with means to display the frequency spectrum of the acousticreflection coefficients as a function of borehole depth.

The novel features of the present invention are set forth withparticularity in the appended claims. The operation together withfurther objects and advantages of the invention may be best understoodby way of illustration and examples of certain embodiments when taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a schematic threedimensional view illustrating the cylindrical geometry of thesonde-boreholecasing system;

FIG. 2 is a schematic cross section view illustrating the cylindricallayers of the system of FIG. 1;

FIG. 3 is a typical curve of the reflection coefficient frequencyspectrum of a system;

FIG. 4 is a reflection coefficient frequency spectrum showing thevariance of the reflection coefficient with formation type and cement orno cement in the annulus;

FIG. 5 shows multiple reflection coefficient frequency spectra forlarger annuli than those of FIG. 4;

FIG. 6 is a perspective view showing the biradially mounted receivingtransducers used in a zero spacing configuration in the presentinvention along with an acoustic transmitting transducer;

FIG. 7 is a top view showing the biradially mounted segmented receiverpairs of the present invention together with their absorbent acousticbacking material;

FIG. 8 is a schematic illustration showing the well logging tool of thepresent invention suspended in a well bore;

FIG. 9 is a block diagram showing the circuitry of the logging system ofthe present invention;

FIG. 10 is a block diagram showing the surface signal processingcircuitry of the present invention;

FIG. 11 is a schematic timing diagram showing the sequence of acousticsignals arriving at the surface signal processing circuits;

FIG. 12 shows the frequency response of a typical acoustic loggingtransducer with the plurality of narrow bandpass filters of the presentinvention superimposed upon it;

FIG. 13 is a schematic representation .showing a possible well logformat which could be used with the methods and apparatus of the presentinvention.

In a copending patent application entitled, Cement Evaluation Logging by.l. H. Moran et al., Ser. No. 840,290, filed July 9, 1969 which isassigned to the assignee of the present invention, there is describedmethod and apparatus for cement evaluation logging which is based on theconcept of examining certain parameters associated with thecharacteristic frequencies, radial modes or resonances of thewell-bore-casing-annulus-formation system. In the present invention thisconcept is also utilized. The method and apparatus of the presentinvention are directed to determining the cement conditions in casedboreholes by the logging of what may be referred to as the acousticreflection coefficient frequency spectra of the system. The reflectioncoefficients are complex mathematical functions which will besubsequently described. The reflection coefficients are determined bycomparing the relative amplitude and phase of reflected sonic energyimpinging upon paired, biradially mounted, segmented, acoustic receivingtransducers at a plurality of frequencies. Once the acoustic reflectioncoefficients are determined, principles of the present invention may beutilized to determine the cementing effectiveness in thecasing-formation annulus from the log of the reflection coefficients.

Using the cylindrical coordinate system illustrated in FIG. 1, havingcylindrical coordinates r, 0 and z, the displacements of particles ofthe various media in the system when excited by accoustic energy can bedenoted by the functions U,, U. and U Considering the case ofcylindrical symmetry, in which the motion is uniform in all 6, and the zdependence may be ignored, (as discussed in the above referencedcopending Moran et al. patent application, this is valid near thecharacteristic frequencies or modes of the system) only the r dependentmotion need be considered. This type of motion can be written in termsof a scalar potential function :1) defined by U,.= alar The waveequation for the r dependent motion is then 1 Fa Er? s 1 Where c is thespeed of compressional waves in the system. This equation is derivedfrom wellknown physical principles and may, for example, be found byreference to Elastic Waves in Layered Media, by Ewing, Jardetsky andPress, published by the McGraw Hill Company of New York, 1957 Edition.

The scalar potential function d) is, of course, time dependent as wellas dependent on r, [i.e., (r,t)]. However, the time dependence may betaken to be that of a travelling wave [i.e., (r,t) F (r) e '"]where w= 21; f, is the angular frequency of the wave motion of frequency f. Usingthis relation, Equation (1) can be brought into the form Equation (2)isrecognized as a form of Bessels equation having a general solution whichmay be writte n in the form F(r) A H (kr) B H Ucr) g l-lere k rule isthe wave number of the waves. H "(kr) and H (kr) are called Hankelfunctions of the first and second kind, zero order. The coefficients Aand B may be thought of as the complex amplitudes of incoming andoutgoing waves respectively, the As being the amplitude of incomingwaves and the Bs as the amplitude of outgoing waves.

Recalling the definition of the displacement U, in terms of thepotential function (the time dependent exponential function beingdropped since it is merely a phase factor which does not influence theamplitude of the waves) the displacement may be written where A and p.are the Lameconstants which define the elastic properties of anisotropic solid. These constants are related to the compressional C andshear C wave velocities in a solid by the following equations where p isthe density of the material. Using these relations the stress may bewritten as The general solution (Equation 2a) above is valid in each ofthe several layers. This solution may now be adapted to the case of aparticular problem of interest such as the cement evaluation problem.This may be accomplished by applying the boundary conditions of theproblem at hand to determine the complex coeffcients A and B.

First (referring to FIG. 2) there are five layers (numbers 1-5) involvedin the cement evaluation problem. While the general solution is valid ineach layer, the coefficients A and B will be different in each layer.These may be referred to as A and B, for their respective layers. Ingeneral if n is the number of layers there will be 2n unknown complexcoefficients to determine by applying the boundary conditions. Twoboundary conditions which may be applied are l. The displacement U,- iscontinuous at the boundaries between each of the layers, i.e.,

2. The normal stress 1- is continuous at the boundaries between each ofthe layers, i.e.,

Since there are n-l boundaries between n layers of material the twoboundary conditions will furnish 2 (n-l or 2n-2 equations in the Zncoefficients. Hence, two further boundary conditions must be imposed onthe system to obtain a complete solution. These are supplied in thefollowing manner.

In the outer layer (the formation layer 1) there is no further outerboundary to reflect waves. Hence the coefficient A for the outer, layermust be zero (i.e., since the As may be thought of as the amplitudes ofincoming waves). Also, inside the sonde (layer 5) it is impossible todistinguish incoming from outgoing waves at the origin of the coordinatesystem since, as the waves pass through the origin, they change fromincoming waves to outgoing waves. This means that the coefficient A mustequal 8,. These two extra boundary conditions provide the needed 2nequations which may be solved for the 2n unknown complex coefficients Aand B.

Finally, to complete the solution of the Zn equations for the As and Bsan account of the acoustic energy source used to excite the fluid mustbe provided. This represents the acoustic signal coming from the sondeand can have a frequency response comparable to that of typical acousticlogging transducers. This source function can be a function ofdisplacement which is added to the boundary condition for displacementat the sonde-mud (i.e., layer 4-5 of FIG. 2) interface. A function ofthe form With these boundary conditions, the Zn simultaneous equationsfor the coefficients A, and B, can be solved and various properties ofthe system which may be of interest can be observed. For example, onequantity, the ratio AJB, has been found to be of particular interest inevaluating cement conditions in the annulus between the casing and theformation. This quantity represents the ratio of the amplitudes ofincoming to outgoing waves in the mud inside the casing. Such a quantityis accessible to measurement and may be thought of as a reflectioncoefficient which indicates how much acoustic energy escapes in theradial direction from the system into the formation. I

FIG. 3 shows a representative reflection coefficient curve. Thereflection coefficient Q is plotted vs. frequency for the values givenin Table I. A value of reflection coefficient near 1 indicates verylittle energy escaping radially into the formation. Lower valuesindicate more energy escaping into the formation at a particularfrequency. The first minimum 31 of FIG. 3 is due to a transmissionresonance of the casing-annulus system at about 9KI-Iz. The secondminimum 32 is due to a transmission resonance of the annulus itself atabout 27KHz. This frequency corresponds to a wavelength which is equalto twice the annulus width. Other minima such as 33 appear at harmonicsof this latter frequency for the wavelengths 2 [In n l, 2, etc., where lis the annulus width. An approximate expression for these frequenciesis:

where c is the speed of sound (compressional) in the annulus. From thisit is apparent that if the annulus material is changed (i.e., cement orno cement) that the frequencies of the reflection coefficient minima arecorrespondingly changed.

Uncemented (Fluid Filled) Annulus The graphs of FIG. 4 clearlyillustrate this point. The graphs are for the borehole parameters ofTable II for both uncemented annulus and cemented annulus, bounded byboth hard and soft formations. The graphs of FIG. 4 show that the TABLEII Casing Outer Diameter 6 inches Casing Thickness .25 inches BoreholeDiameter 8 inches Formation Compressional Travel Time (soft) I 4. SEC/FT(hard) 50 p. SEC/FT reflection coefficients curves have greatdifferences between the cemented and uncemented cases, particularly inthe frequency range just above the first minimum of the curves (i.e.,from ISKl-Iz to 20KHz). These differences have been found to appearindependently of the spacing between the acoustic transmitter andreceiver on the sonde and over a wide range of small annulus(micro-annulus) sizes between the casing and the cement. A quantativemeasure of these differences can be made which would be an accurateindicator of cement conditions in the annulus. The method and apparatusto be discussed are intended to make just such a measurement.

If only the frequency range just above the first minimum in thereflection coefficient curve is used, however, the results may not bereliable. This is due to the additional effect illustrated in FIG. 5.The graphs of FIG. 5 correspond to the parameters given in Table III.

Formation Compressional Travel Time (hard) 50 p. SEC/FT (soft) [00 p.SEC/FT These graphs illustrate the effect of larger annulus widths onthe reflection coefficient curves for cemented and uncementedconditions. As may be appreciated from these plots, for the largerannulus width the second minimum for the uncemented case (51 52 on FIG.5) is shifted lower in frequency and, in general, tends to produceanomalous results in the frequency range 20 to 25 KHz. That is, contraryto the general trend, the uncemented conditions generally result inlower reflection coefficients than those corresponding to the cementedconditions in this frequency range. For still larger annulus widths thanthose of the cases illustrated in FIG. 5, the third and higher minimacan intrude into the low frequency range and further disturb the resultsexpected from curves such as those of F IG. 4. Since the annulus widthmay not be accurately known in all cases of actual well bores (i.e.,generally the annulus width at a location will not be known unless acaliper is run prior to the setting of the casing and even then thecementing process can change the borehole diameter due to washouts orthe like) some means must be provided to measure the cement in theannulus which is independent of the annulus width.

From the above discussion it will be appreciated that it would not bedesirable to rely on one measurement of the minima of the reflectioncoefficient curve for a system to establish the cementing effectiveness.A proposal to measure a quantity which is proportional to the reflectioncoefficient in three relatively wide frequency bands in order to avoidthe above-mentioned anomaly is described in a copending applicationentitled, Cement Evaluation Logging Utilizing Reflection Coefficients"Ser. No. 840,335, filed July 9, 1969 by Ralph G. Biel, which is assignedto the assignee of the present invention. The present invention,however, proposes to evaluate the cementing effectiveness by logging thereflection coefficients at a multiplicity of frequencies over theacoustic spectrum at each selected borehole depth. The resultingreflection coefficient frequency spectra may then be utilized todetermine cementing effectiveness.

boundary of an angularly segmented receiving trans- 5 ducer in the mudinside the casing may be written as a function of frequency as Pli(w)Ai(u) H "(Kre) Bi(w)H (Kre) 9 Here, I l, 2-denotes the particularreceiver in question, i represents the ith angular segment of thatreceiver, r, is the radius of the ith receiver and H (kn) and H (kn) areagain the Hankel function of the first and second kinds, zero order.Also, m=21rf is again the angular frequency of the waves and k= rule is,again, the wave number of the waves. The functions A, (u) and B. (u) arecomplex coefficients (functions of the frequency,m) differing only by amultiplicative constant (L k')\) from the A's and 8's previously 2discussed.

Solving Equation (9) for the As and B's and taking the ratio of these attwo receivers of different radii r and r, (here the multiplicativefactors L =k)t cancel making the A's and Bs the same as previouslydiscussed) the acoustic reflection coefficient Q BIA may be writtenexactly as recejver radii r, and r, as

3201 (a) am) where the functions p(w) and 6(a)) are the amplitude andphase respectively of the ratio of the complex Hankel functions. Thisquantity is constant and known for a particular frequency and hence, maybe precomputed and stored, for example, as a voltage level, for lateruse in calculating the reflection coefficients.

In terms of quantities which may be observed at the surface of twotransducers of different radii inside the casing, the amplitude and thephase of reflected acoustic waves may be observed. These quantities willbe denoted by p and it respectively. Thus, by using the relationship ofEquation (ll), the more complex Equation (10) may be brought into themore usable form for the magnitude of the reflection coefficient |Q|[Pi-P2P os (0+1=)]+[m si +.-.)1*

Here the i subscript has been dropped for brevity of expression but itshould be understood that Equation l2) and the succeeding equationsapply equally well to segmented receiver transducers having 1' segments.

Suitable algebraic manipulation of Equation l2)can be made to yield anexpression for IQI in terms of the relative amplitudes p lp R and phasesd of the reflected waves at the transducer boundaries. This yields:

The quantities i and abbi'e are'm relative V amplitude and phase at thetwo receivers of radii r, and

r, and, if measured, can be combined as indicated in Equation (l3)togive the acoustic reflection coefficient at a particular frequency(recalling thatp and 0 are known for each frequency once the receiverradii are specified). Thus, a plurality of measurements of R and ti: maybe made at different frequencies over the spectrum of acoustic energyused to excite the formation and, with appropriate processing, thesewould yield logs similar to the graphs of FIGS. 3, 4 and 5 which may beused to interpret the cement conditions behind the casing. The apparatusto be described is intended to provide just such measurements andcomputations, enabling the production of reflection coefficient curvesfor this purpose.

Referring now to FIG. 6, a transducer configuration to obtain areflection coefficient log in accordance with the present invention isshown schematically. A hollow cylindrical piezoelectric transmittingtransducer 61 of conventional design may be used to produce the acousticenergy used to excite the system. Just above and below the transmittingtransducer 61 in a zero spacing arrangement are placed acousticreceivers 62 and 63 which may also use conventional piezoelectricmaterials to sense impinging reflected acoustic energy and produce anelectrical output representative thereof. Receivers 62 and 63, it shouldbe noted, are of different radiir, and r and are comprised of aplurality of planar angular segments of piezoelectric material 62a and63a assembled into a roughly cylindrical shape. Alternatively receiversegments 62a 63a could be of an arcuate shape, even more closelyapproximating a cylindrical surface.

As may be more readily appreciated in the top view of the receivingtransducer assembly (with some segments omitted for clarity) shown inFIG. 7, the individual angular segments of the receiving transducers arearranged about a cylindrical backing of acoustic absorbent base material73 and 74. For example, the angular segments 72 of the receivingtransducer having the smaller radius r, may be cemented or otherwiseconveniently mounted about the exterior surface of a cylindrical shapedpiece of acoustic absorbent material such as that marketed under thetrade name MINK-2000 which is available from the Johns-ManvilleCorporation. Similarly angular receiver segments 71 of the largerreceiving array can be mounted about the exterior of another cylindricalshaped piece of acoustic absorbent material 74. It should be noted thatboth receivers 62 and 63 have the same number of angular segments in aone-to-one relationship. These may be referred to as paired receiversegments. This enables the relative amplitude p,/p, and the relativephase over the same angular arc of the borehole to be observed by thecorresponding paired receiver segments of each receiver array. Since thereceivers are backed by an acoustically absorbent material, eachreceiver segment, to a good approximation, sees only acoustic energyreflected from the angular portion of the borehole-casing-formationsystem which is directly in front of it. This type of mounting of pairedreceiver segments on two cylinders of different radii will henceforth bereferred to as biradial mounting of the receiver segment pairs.

FIG. 8 shows an acoustic well logging tool 81 in accordance with theprinciples of the present invention suspended from a winch arrangement88, as well known in the art, by a multiconductor cable 82 in afluid-filled well bore 83. Well bore 83 has been lined with casing 84,cemented in place by a cement layer 85 whose condition is to bedetermined by means of obtaining acoustic reflection coefficient logs aspreviously described. It will be understood that cable 82 contains allnecessary conductors for providing power and control signals to thedownhole tool 81 and additionally provides a communication path to thesurface equipment for electrical signals representative of acousticenergy received by the receiving transducers 86 and 87 of the downholetool.

Logging tool 81 contains an acoustic transmitter 89 and a pair ofsegmented biradially mounted acoustic receivers 86 and 87 as justdescribed. While only one such transducer array is shown it will beunderstood that a plurality of such arrays may be used if desired, inorder to utilize the same logging tool for other measurements such asacoustic travel time or attenuation logs. The upper portion of tool 81houses appropriate control circuitry 90 which, in conjunction withsurface control circuits 91, operate to control the measuring sequencefor obtaining the reflection coefficient logs. These circuits actgenerally, as will be described in more detail subsequently, to fire theacoustic transmitter 89 and then to sample the resulting outputs fromthe biradially mounted, segmented receiver pairs. The receiver pairoutputs are sampled over the same time interval following thetransmitter firing but are sequentially transmitted to the surface forprocessing by the signal processing circuits.

This may be more readily appreciated by reference to FIG. 9 which showsa block diagram of the control circuitry. As sequential processing ofsignals at the surface and downhole is involved, it will be appreciatedthat synchronization of the process must be established. Suchsynchronization is supplied by the 60 per second AC power line referencefrequency 101 which is accurate and readily available. By using a commonreference frequency both downhole and on the surface and generatingtiming pulses at the surface and in the downhole tool based onsubmultiples such as one-half, one-third or any other desired fractionof the reference frequency, the synchronization is achieved. Timingpulse generators 102 and 103 which may be of the type shown in moredetail in US. Pat. No. 3,304,537 may be utilized for this purpose.

In order to provide a smooth and continuous appearing log it isdesirable to operate the acoustic transmitter in a pulsed mode ofoperation using as high a repetition frequency as possible. For example,a transmitter firing frequency of 10 times per second could provide thiseffect satisfactorily, thus giving a measurement cycle of 100milliseconds duration. The uphole sequencer 104 provides a reset pulseevery milliseconds on line 106 to start the measuring sequence in thedownhole tool. This pulse is also used to reset all circuits in thesignal processing circuits 107 to enable a new series of computationsfor the new measurement cycle to be begun in the uphole signalprocessing circuitry.

Upon receipt of the reset pulse, the downhole sequencing circuit 108,after an appropriate delay, generates a transmitter fire pulse on line109 which is used to fire the acoustic transmitter T in a conventionalmanner. Additionally, the transmitter fire pulse is routed to amplifiers110 and 111 where it goes immediately to the surface on both uplinkconductors 112 and 113. This pulse is used to control the signal inputsfrom the downhole tool to the signal processing circuits as will besubsequently described. Once the transmitter is fired, a brief delay isdesirable before activating the receivers in order to allow any ringingor resonance of the transmitter to damp out. The sequencer 108 providesthis delay by not supplying conditioning pulses to signal gates 114 and115 until after the desired delay time has elapsed. Signal gates 114 and115 will permit no signals from the commutators 116 and 117 to enteramplifiers 110 and 111 for transmission to the surface until conditionedby the sequencer conditioning pulses just mentioned.

At the same time that the signal gates 114 and 115 are conditioned, thesequencer 108 also supplies a pulse on line 119 to enable the commutatordriver 121 to receive power from power amplifier 120 to start thecommutators 116 and 117. Power amplifier 120 amplifies the timing pulsesprovided by the timing pulse generator 103 to power the commutatordriver 121 which may be a synchronous motor or the like. Since thecommutator driver 121 is powered in this manner its synchronization withthe remainder of the system is assured.

Once the commutator is started, the sampling of reflected acousticsignals impinging on the two receiver arrays R, and R is begun. Pairedbiradially mounted receiver segments of R, and R are sampledsimultaneously by the action of ganged commutators 116 and 117. Althoughonly four pairs of receiver segments are shown for simplicity, it willbe understood that as many pairs as desired may be used.

Each pair of corresponding receiver segment outputs are delayed by thesame amount and appear at their appropriate commutator contacts foramplification and transmission to the surface simultaneously. Forexample, delay lines 122 and 123 which delay the outputs of pairedbiradially mounted receiver segments R and R each have the same delaytime, denoted by D,. Thus, the ganged commutators 116 and 117 sampleoutputs of R and R which occurred over the same time period. The delaytimes D,, D,, D, and D, provided by the respective delay lines are soarranged as to make the signals appearing at the commutator inputs whenthe commutator wiper is there to sample them, all have the same originin time. That is to say, the delay provided by delay lines 124 and 125,(i.e., D is equal to D, plus the time it takes the commutator to movefrom the contacts sampling the inputs from paired segments A to thecontacts sampling the inputs from paired segments B of the receivers R,and R The result of this is to produce a sequence of receiver outputs onthe uplink lines 112 and 113 such as those appearing in FIG. 11.

In FIG. 11, the signals appearing on the uplink line 112 is illustrated.At the beginning of the measurement cycle the t, or transmitter firepulse is generated as previously described and appears first on theuplink. After the ringing delay, the output of one of the pairedbiradially mounted receiver segments, say R appears, followedsequentially in time by the outputs of paired segments R R and R,,,. Allof these receiver outputs have the same origin in time, as previouslydiscussed, but appear sequentially on the uplink lines 112 and 113 forprocessing by the surface signal processing circuits.

Referring now to FIG. 10, the operation of the surface signal processingcircuits may be more readily appreciated. Recalling the sequence ofsonic receiver signals from the downhole tool coming to the surface onuplink lines 112 and 113, the transmitter fire, or t signal whichprecedes the receiver signals is used to condition the surface signalprocessing circuits for the computational cycle. A t, detector 141detects the arrival of the t, pulses on both lines 112 and 113 and whenboth are detected produces an output pulse to the "ringing" delayone-shot 142. Detection of the t, pulse on both lines provides a measureof redundancy to the system for insuring that all downhole circuits areproperly functioning. Delay one-shot 142 upon receipt of the I, detectpulse is in its zero output or reset condition due to its prior receiptof the reset pulse from the sequencer 104 of FIG. 9 as previouslydiscussed. When the I, detect pulse arrives, delay one-shot 142 delaysfor a period sufficient for the transmitter ringing" to die out and thenproduces an output pulse on line 143 to condition signal input gates 145and 146 to permit entry of the receiver signals into the signalprocessing circuits. The output of delay one-shot 142 continues (i.e.,gates 14S and 146 remain open) until the delay one-shot 142 receives areset pulse from the sequencer 104 at the end of the computation cycle.By keeping the signal input gates closed during the firing oftransmitter and between computational cycles, the possibility of noiseor spurious cross-talk signals entering the system is minimized.

Upon entry into the signal processing circuits the sequentially arrivingpaired receiver signals are amplified by amplifiers 147 and 148 and theninput into two circuits 149 and 150 each consisting of a plurality ofrelatively narrow bandwidth fixed bandpass frequency filters (referredto hereafter as filter banks"). These bandpass filters break up thepaired receiver inputs into a plurality of discrete frequency componentswhich are sequentially processed as will be described subsequently inorder to compute the reflection coefficient Q in the range of each ofthe bandpsss filters. Each bandpass filter in the group may have, forexample, a half power bandwidth of 3KI-Iz, and enough of the filters areutilized to cover the frequency range of interest for logging Q. Forexample, 20 such filters would cover the range from to 60I(I-Iz, insuccessive 3KI-Iz bands. Thus, it may be said that the bandpass filters149 and 150 perform a Fourier analysis of the input waveforms of thepaired receiver segments arriving on the input lines 112 and 113,breaking them up into a plurality of discrete frequency components F toF and F to F as illustrated in FIG. 10.

Recalling that an individual frequency component of the signal may bethought of as the product of an amplitude times a phase factor [i.e., F(m) p] the next step in the signal processing is to extract theamplitude and phase information from the discrete frequency componentscomprising the outputs of the'bandpass filter banks 149 and 150. Indiscussing the further signal processing it should be noted that eachbank of bandpass filters 149 and produces a plurality of outputs. Forease of discussion only one pair of outputs (those labeled F and F inFIG. 10) will be described. It will be understood that the remainingoutputs F to F and F to F are processed in the same manner.

Before the amplitude and phase data can be separated from the filteroutputs, any distortion due to the frequency response of the receivingtransducers must be removed. This function is performed by variable gainamplifiers 151 and 152. This concept may be more readily appreciated byreference to FIG. 12 which illustrates the frequency response of atypical piezoelectric receiving transducer to a constant amplitudesignal swept across the frequency spectrum. The passbands of a pluralityof bandpass filters F, to F are shown superimposed on this responsecurve. It will be noted that such a transducer is considerably moresensitive in the frequency region of filters F and F say, in the regionof filters F b or F,. Thus, the outputs of filters F, or F, would beamplified by their associated variable gain amplifiers, corresponding tovariable gain amplifiers 151 and 152 more than the outputs of filters Fand F Once the frequency response of the receiving transducers is knownand the bandwidth and center frequency of the filters is established,the gain of variable amplifiers 151 and 152 and their correspondingcomponents in the outputs of the other filters may be adjusted toperform this connection. Such a process may be referred to asnormalizing the outputs of the filters.

In order to extract the relative phase factor it, da and the relativeamplitude ratio p lp which are needed to compute the reflectioncoefficient Q in a particular frequency band, the normalized outputs ofthe variable gain amplifiers 151 and 152 are input to a phase comparatorcircuit 153 and an amplitude divider circuit 154. Phase comparatorcircuit 153, which may be a phase detector such as a discriminatorcircuit or the like, produces an output voltage which is proportional tothe phase difference mp 41 of its two inputs. The amplitude dividercircuit 154 furnishes an output voltage which is proportional to theamplitude ratio of its two inputs p lp Similarly, the normalized outputsof each of the paired receiver segment signals from the remainder of thebandpass filters F to F, and F to F are input to corresponding phasecomparators and amplitude divider circuits (not shown) which extract thecorresponding relative phase ((11 4: and amplitude ratio (p /p data.

The outputs of the plurality of phase comparators and amplitude dividersfor each pair of matching bandpass filters in filter banks 149 and 150are input to a multiplex stage 155. Mutliplex stage 155 functions todelay for variable intervals the outputs of the phase comparators andamplitude dividers and to present these phase differences, b to dz andamplitude ratios p lp to p lp sequentially as inputs to the reflectioncoefficient computer 156 so that the computer can calculate thereflection coefficients for each of the filter bands F, to F,,. To thisend timing pulses are presented to the multiplex stage from the timingpulse generator 102 of FIG. 9. A plurality of delay lines (not shown) inthe multiplex stage could, for example, be gated sequentially by aplurality of gates conditioned from a counter (not shown) operated bythe timing pulses input to the multiplex stage. The multiplex stagecould then, as the counter reached a certain threshold count for aparticular filter pair, say F and F gate the corresponding phasedifference and amplitude ratio for this filter pair into the reflectioncoefficient computer. This process would be repeated for each filterpair F and F until all pairs were finished. The counter could then berecycled to operate on the filter outputs from the next receiver segmentsignals which would be arriving on the uplink input lines.

The reflection coefficient computer 156 receives the relative phasedifferences 42 4),, and p lp R, and uses these in Equation (13) tocompute the corresponding reflection coefficient Q for the particularfrequency band in question. It will be recalled from Equation (1 I) thatthe constants p and of Equation (l3) are known for a particularfrequency once the radii r and r of the two receiver segments are known.Hence, all the quantities on the right hand side of Equation (13) areknown and the computer 156 performs the indicated operations of Equation(13) to compute the magnitude of the reflection coefficient Q. It shouldbe noted that in using this technique to compute the reflectioncoefficients IQI that no mathematical approximations have been made.Thus, the computed output of the reflection coefficient computer isexactly the cylindrical reflection coefficient.

The reflection coefficients Q to Q, for a particular pair of biradiallymounted receiver segments are thus sequentially output from computer156. The sequentially output Qs may then be routed to a de-multiplexstage 157 for display purposes. De-multiplex stage 157 performs theinverse function of the multiplex stage 155. That is to say, thede-multiplex stage 157 takes the sequentially arriving Q and delays themby varying amounts so as to output all the Q's for the spectrum of aparticular receiver simultaneously. This enables a multi-channel displaydevice to display the reflection coefficient frequency spectrum of aparticular receiver segment pair.

For example, a cathode ray tube variable density display could be madeof the reflection coefficient spectrum as a function or borehole depthfor each of the paired receiver segments of the receiving transducers inthe downhole tool. If there were three such pairs of receivingtransducers then three channels on the display would be required. Thesethree channels could be placed side by side on the recording medium inthe manner shown in FIG. 13. Interpretation of the reflectioncoefficient spectra of the different receiver pairs would then yieldvaluable information as to the cement conditions in each of the fourquadrants of the borehole in front of each pair of biradially mountedreceiver segments.

Alternatively, the sequential Q, output of the reflection coefficientcomputer could be routed to cement evaluation logic circuitry 158 whichcould be similar to that disclosed in the above-mentioned copending Beilapplication. The cement evaluation logic circuitry could then perform alogical analysis of the reflection coefficient spectra and output asingle cement evaluation log as a function of borehole depth.

Referring now to FIG. 13, one format for displaying the reflectioncoefficient log as a function of borehole depth is illustratedschematically. In the illustration of FIG. 13 a cathode ray tubevariable density display for three segmented biradially mounted receiverpairs labeled R., R, and R is shown on the left-hand side of the logwhile on the right-hand side a cement evaluation log for each of thethree receiver pairs is displayed. The cement evaluation log could be,as previously discussed, the result of applying cement evaluation logicby a computer 158 of FIG. 10 to the reflection coefficient data. Theresult of this logic would be to produce a single trace as a function ofborehole depth of the cement quality or effectiveness for each receiverpair. This curve could range between 0 and percent cement effectivenessor the like.

The variable density cathode ray tube display on the left illustratesthe inverse of the magnitude of the reflection coefficients seen by eachreceiver pair as a function of frequency, the frequency increasing tothe right of the illustration. The cathode ray tube beam in such adisplay is intensity modulated in proportion to the magnitude of thedisplayed quantity. The schematic display shown, for example, might berepresentative of a reflection coefficient spectra such as that shown inFIG. 3. The darker areas of the display would correspond to thereflection coefficient minima 31, 32 and 33 of FIG. 3 at the frequenciesindicated. A display such as that shown in FIG. 13 has the advantage ofshowing simultaneously the reflection coefficient spectrum and cementevaluation log for each receiver pair used. It will be recalled thateach receiver pair to a good approximation only sees that portion of theborehole-casing-annulus-formation system which is directly in front ofit because of its acoustically absorbent mounting. Thus, a display suchas that of FIG. 13 gives a complete picture of cement conditions in theangular arc of the case borehole before each receiver. While only threesuch displays are shown in FIG. 13, it will, of course, be appreciatedby those skilled in the art that as many such segmented receiver pairscould be used as desired to give finer resolution of the angulardisposition of the cement, if desired.

Other alternative ways of processing the reflection coefficient datawill be apparent to those skilled in the art. For example, thereflection coefficient data could be recorded on magnetic tape andprocessed at a remote site by a general purpose digital computer. Or, ifa small general purpose digital computer were available at the wellsite, it could be used to accept the outputs of the multiplex stage andcould compute the reflection coefficient data internally and eitherdisplay it as a multi-channel variable density display as previouslydiscussed or apply cement evaluation logic and display a cementevaluation log. These and other alternatives such as magnetic taperecording the multiplex stage outputs for later processing, or recordingthe raw sequential receiver segment waveforms for remote processing in amanner analogous to that of the apparatus disclosed are all consideredto be within the scope of the invention.

While a particular embodiment of the present invention has been shownand described, it is apparent that changes and modifications may be madewithout departing from this invention in its broader aspects; and,therefore, the aim in the appended claims is to cover all such changesand modifications as fall within the true spirit and scope of thisinvention.

What is claimed is:

l. A method of acoustically logging a well bore to provide a frequencyspectrum of the acoustic reflection coefficient comprising the steps of:

transmitting a component of acoustic energy radially outwardly from theborehole into the cement and surrounding formations;

receiving a reflected portion of the transmitted acoustic energy at twodifferent radii in the borehole and generating first and secondelectrical signals representative thereof;

passing said first and second signals through n, where n is an integernumber, relatively narrow bandpass filters having different centerfrequencies spaced over the acoustic spectrum to provide it pairs ofcorresponding frequency components for said first and second signals;

generating third electrical signals representative of the relative phaseof each of said n pairs of corresponding frequency components;

generating fourth electrical signals representative of the relativeamplitudes of each of said n pairs of corresponding frequencycomponents; and

combining said third and fourth signals to generate n signalsrepresentative of the acoustic reflection coefficient for each of the nfrequency components, thereby providing a frequency spectrum of acousticreflection coefficients.

2. The method of claim 1 and further including the step of recordingsaid n signals representative of the acoustic reflection coefficient foreach of the n frequency components.

3. The method of claim 1 wherein the steps are performed periodicallywhile moving a logging tool vertically through the well bore and thefrequency spectrum of acoustic reflection coefficients is recorded as afunction of borehole depth at each level traversed by the logging tool,

4. A method of acoustically logging a well bore to provide a frequencyspectrum of the acoustic reflection coefficient comprising the steps of:

transmitting a component of acoustic energy radially outwardly from theborehole into the cement and surrounding formations;

receiving a reflected portion of the transmitted acoustic energy at apair of biradially mounted acoustic receivers, each having I angularsegments, where l is an integer number 2 or greater, and generating Ipairs of electrical signals representative of the received acousticenergy at each corresponding pair of receiver segments;

passing each of said 1 pairs of corresponding signals through n, where nis an integer number, relatively narrow bandpass filters havingdifferent center frequencies spaced over the acoustic spectrum toprovide n pairs of corresponding frequency components for each of the 1corresponding receiver segment pairs;

generating electrical signals representative of the relative phase ofeach of said n pairs of corresponding frequency components for each ofsaid 1 corresponding receiver segment pairs;

generating electrical signals representative of the relative amplitudesof each of said n pairs of corresponding frequency components for eachof said 1 corresponding receiver segment pairs; and

combining said relative phase and relative amplitude signals for each ofsaid-n pairs of corresponding frequency components for each of said Icorresponding receiver segment pairs to generate n signalsrepresentative of the acoustic reflection coefficient for each of the nfrequency components thereby providing 1 frequency spectra of acousticreflection coefficients corresponding to the reflection coefficientspectrum of each of the I angular segments of the biradially mountedacoustic receivers. 5. The method of claim 4 and further including thestep of recording said n signals representative of the acousticreflection coefficients for each of the n frequency components and sodoing for each of the 1 frequency spectra of acoustic reflectioncoefficients corresponding to the reflection coefficient spectrum ofeach of the l angular segments of the biradially mounted acousticreceivers.

6. The method of claim 4 wherein the steps are performed periodicallywhile moving a logging tool through the well bore and the resulting 1frequency spectra of acoustic reflection coefficients are recorded as afunction of borehole depth at each level traversed by the logging tool.

7. A method for acoustically logging a cased borehole to determine theeffectiveness of the cement disposed in the annulus between the casingand the formations comprising the steps of:

moving an acoustic well logging tool having at least one acoustictransmitter and at least two acoustic receivers of different radiithrough the well bore;

periodically firing said acoustic transmitter to propagate a componentof acoustic energy radially outwardly from the borehole into the cementand surrounding formations; periodically activating said acousticreceivers in response to said transmitter firing to receive a reflectedportion of the transmitted acoustic energy at, at least two differentradii in the borehole and generating a plurality of electrical signalsrepresentative of said reflected portion of acoustic energy; separatingeach of said representative electrical signals into a plurality ofsignals representative of discrete frequency components, the frequencycomponents of each representative signal being in a one-to-onecorrespondence with each other;

generating electrical signals representative of the relative phases ofcorresponding pairs of said frequency component signals;

generating electrical signals representative of the relative amplitudesof corresponding pairs of said frequency component signals; and

combining said relative phase signals and said relative amplitudesignals for each of the corresponding pairs of said frequency componentsignals to 2. generate signals representative of the acoustic reflectioncoefficient in the frequency region of each of said discrete frequencycomponents.

8. The method of claim 7 and further including the step of recording asa function of the borehole depth of said logging tool, said signalsrepresentative of the acoustic reflection coefficient in the frequencyregion of each of said discrete frequency components.

9. The method of claim 7 wherein the step of breaking each of saidrepresentative electrical signals down into a plurality of signalsrepresentative of discrete frequency components is performed by applyingeach of said representative signals to a plurality of relatively narrowbandpass frequency filters having different center frequencies disposedover a selected portion of the acoustic spectrum.

10. The method of claim 7 wherein said acoustic receivers are angularlysegmented and have receiver segment pairs in one to one correspondencewith each other, each corresponding receiver segment pair disposedadjacent the same angular arc of the borehole and the remaining methodsteps pertaining to the signal processing are each performed on therepresentative signals generated by each corresponding receiver segmentpair, thereby providing an independent frequency spectrum of acousticreflection coefficients for each angular arc of the borehole.

11. The method of claim 7 and further including the step of correctingsaid signals representative of discrete frequency components of therepresentative signals for any amplitude distortion introduced by thefrequency response of the acoustic receivers by amplifying each signalrepresentative of a discrete frequency component by an amount inverselyproportional to the sensitivity of the acoustic receiver in thefrequency region said signal represents.

12. A method of acoustically logging a cased borehole with alongitudinally extending exploring device to provide a plurality offrequency spectra of acoustic reflection coefficients comprising thesteps of:

transmitting a component of acoustic energy radially outwardly from theexploring device into the cement and surrounding formations; receiving areflected portion of the transmitted acoustic energy at two differentpositions within the borehole which are at two positions havingdifferent radii relative to the central axis of the exploring device andgenerating first and second electrical signals representative thereof;

separating said first and second signals into a plurality ofcorresponding pairs of frequency component signals; and

determining from said plurality of corresponding pairs of frequencycomponent signals, the plurality of acoustic reflection coefficientscorresponding to each of the pairs of frequency component signals, andgenerating signals representative thereof.

13. The method of claim 12 and further including the step of recordingsaid plurality signals representative of the acoustic reflectioncoefficient corresponding to each of the frequency component signals.

14. The method of claim 12 wherein the steps are performed periodicallyas a well too] is moved through the borehole and the signals arerecorded as a function of borehole depth.

15. Apparatus for acoustically logging a cased well bore to provide anacoustic reflection coefficient spectrum comprising:

an acoustic well logging tool having at least one acoustic transmitterand at least two acoustic receivers of different radii;

means for periodically firing said transmitter to propagate a componentof acoustic energy radially outwardly from the borehole into the cementand surrounding formations; means for periodically activating saidacoustic receivers in response to said transmitter firings to receive areflected portion of the transmitted energy at said different radii andto generate a plurality of electrical signals representative thereof;means for separating said plurality of representative electrical signalsinto a plurality of signals representative of discrete frequencycomponents of said signals, said frequency components being in aone-to-one correspondence with each other;

means for generating electrical signals representative of the relativephases of corresponding pairs of said frequency component signals;

means for generating electrical signals representative of the relativeamplitudes of corresponding pairs of said frequency component signals;and

means for combining said relative phase signals and said relativeamplitude signals to generate signals representative of the acousticreflection coefficient in the frequency region of each of said discretefrequency components.

16. The apparatus of claim 15 wherein said means for separating saidplurality of representative electrical signals into signalsrepresentative of discrete frequency components includes a plurality ofrelatively narrow bandpass filters having different center frequenciesdisposed over a selected portion of the acoustic spectrum.

17. The apparatus of claim 15 wherein said means for generating signalsrepresentative of the relative phases of corresponding pairs of saidfrequency component signals includes phase comparator means forcomparing the phases of said signals and means for generating an outputsignal representative of the difference of the phases of said signals.

18. The apparatus of claim 15 wherein said means for generatingelectrical signals representative of the relative amplitudes ofcorresponding pairs of said frequency component signals includes meansfor comparing the ratio of the amplitudes of said signals and means forgenerating an output signal representative of said amplitude ratio.

19. The apparatus of claim 15 wherein said means for combining saidrelative phase and said relative amplitude signals includes means forcombining a signal representative of the difference of the phases ofsaid signals and the ratio of said amplitudes, together with means forgenerating signals representative of the acoustic reflection coefficientderived from the phase difference and amplitude ratio signals.

20. Acoustic apparatus for logging a well bore to provide acousticreflection coefficient logs of the well bore comprising:

acoustic transmitting transducer means;

. first piezoelectric acoustic receiver means of a hollow cylindricalshape having a radius n;

acoustic receiver means are divided into n, where n is an integer number2 or greater, corresponding angular segments, whereby each of the 11corresponding receiver segment pairs receive reflected acoustic energyfrom the angular portion of the borehole in front of the receiversegment pairs.

23. The apparatus of claim wherein said first and second receiver meansare disposed along the longitudinal axis of the transducer array oneither side of the transmitting transducer in a zero spacingarrangement.

24. Apparatus for acoustically logging a cased well borehole todetermine the effectiveness of cement disposed in the annulus betweenthe casing and the formation comprising:

means for transmitting a component of acoustic energy radially outwardlyfrom the borehole into the cement and surrounding formations;

means for receiving a reflected portion of the transmitted acousticenergy at two different radial positions in the borehole and generatingelectrical signals representative thereof;

means for performing a Fourier analysis of said first and secondelectrical signals to separate said signals into a plurality ofcorresponding pairs of frequency component signals; and

means responsive to said plurality of corresponding pairs of frequencycomponent signals for determining the plurality of acoustic reflectioncoefficients corresponding to said frequency component signals andgenerating signals representative thereof.

25. The apparatus of claim 24 and further including means for recordingas a function of borehole depth said signals representative of theacoustic reflection coefficients corresponding to said pairs offrequency component signals.

1. A method of acoustically logging a well bore to provide a frequencyspectrum of the acoustic reflection coefficient comprising the steps of:transmitting a component of acoustic energy radially outwardly from theborehole into the cement and surrounding formations; receiving areflected portion of the transmitted acoustic energy at two differentradii in the borehole and generating first and second electrical signalsrepresentative thereof; passing said first and second signals through n,where n is an integer number, relatively narrow bandpass filters havingdifferent center frequencies spaced over the acoustic spectrum toprovide n pairs of corresponding frequency components for said first andsecond signals; generating third electrical signals representative ofthe relative phase of each of said n pairs of corresponding frequencycomponents; generating fourth electrical signals representative of therelative amplitudes of each of said n pairs of corresponding frequencycomponents; and combining said third and fourth signals to generate nsignals representative of the acoustic reflection coefficient for eachof the n frequency components, thereby providing a frequency spectrum ofacoustic reflection coefficients.
 2. The method of claim 1 and furtherincluding the step of recording said n signals representative of theacoustic reflection coefficient for each of the n frequency components.3. The method of claim 1 wherein the steps are performed periodicallywhile moving a logging tool vertically through the well bore and thefrequency spectrum of acoustic reflection coefficients is recorded as afunction of borehole depth at each level traversed by the logging tool.4. A method of acoustically logging a well bore to provide a frequencyspectrum of the acoustic reflection coefficient comprising the steps of:transmitting a component of acoustic energy radially outwardly from theborehole into the cement and surrounding formations; receiving areflected portion of the transmitted acoustic energy at a pair ofbiradially mounted acoustic receivers, each having l angular segments,where l is an integer number 2 or greater, and generating l pairs ofelectrical signals representative of the received acoustic energy ateach corresponding pair of receiver segments; passing each of said lpairs of corresponding signals through n, where n is an integer number,relatively narrow bandpass filters having different center frequenciesspaced over the acoustic spectrum to provide n pairs of correspondingfrequency components for each of the l corresponding receiver segmentpairs; generating electrical signals representative of the relativephase of each of said n pairs of corresponding frequency components foreach of said l corresponding receiver segment pairs; generatingelectrical signals representative of the relative amplitudes of each ofsaid n pairs of corresponding frequency components for each of said lcorresponding receiver segment pairs; and combining said relative phaseand relative amplitude signals for each of said n pairs of correspondingfrequency components for each of said l corresponding receiver segmentpairs to generate n signals representative of the acoustic reflectioncoefficient for each of the n frequency components thereby providing lfrequency spectra of acoustic reflection coefficients corresponding tothe reflection coefficient spectrum of each of the l angular segments ofthe biradially mounted acoustic receivers.
 5. The method of claim 4 andfurther including the step of recording said n signals representative ofthe acoustic reflection coefficients for each of the n frequencycomponents and so doing for each of the l frequency spectra of acousticreflection coefficients corresponding to the Reflection coefficientspectrum of each of the l angular segments of the biradially mountedacoustic receivers.
 6. The method of claim 4 wherein the steps areperformed periodically while moving a logging tool through the well boreand the resulting l frequency spectra of acoustic reflectioncoefficients are recorded as a function of borehole depth at each leveltraversed by the logging tool.
 7. A method for acoustically logging acased borehole to determine the effectiveness of the cement disposed inthe annulus between the casing and the formations comprising the stepsof: moving an acoustic well logging tool having at least one acoustictransmitter and at least two acoustic receivers of different radiithrough the well bore; periodically firing said acoustic transmitter topropagate a component of acoustic energy radially outwardly from theborehole into the cement and surrounding formations; periodicallyactivating said acoustic receivers in response to said transmitterfiring to receive a reflected portion of the transmitted acoustic energyat, at least two different radii in the borehole and generating aplurality of electrical signals representative of said reflected portionof acoustic energy; separating each of said representative electricalsignals into a plurality of signals representative of discrete frequencycomponents, the frequency components of each representative signal beingin a one-to-one correspondence with each other; generating electricalsignals representative of the relative phases of corresponding pairs ofsaid frequency component signals; generating electrical signalsrepresentative of the relative amplitudes of corresponding pairs of saidfrequency component signals; and combining said relative phase signalsand said relative amplitude signals for each of the corresponding pairsof said frequency component signals to generate signals representativeof the acoustic reflection coefficient in the frequency region of eachof said discrete frequency components.
 8. The method of claim 7 andfurther including the step of recording as a function of the boreholedepth of said logging tool, said signals representative of the acousticreflection coefficient in the frequency region of each of said discretefrequency components.
 9. The method of claim 7 wherein the step ofbreaking each of said representative electrical signals down into aplurality of signals representative of discrete frequency components isperformed by applying each of said representative signals to a pluralityof relatively narrow bandpass frequency filters having different centerfrequencies disposed over a selected portion of the acoustic spectrum.10. The method of claim 7 wherein said acoustic receivers are angularlysegmented and have receiver segment pairs in one to one correspondencewith each other, each corresponding receiver segment pair disposedadjacent the same angular arc of the borehole and the remaining methodsteps pertaining to the signal processing are each performed on therepresentative signals generated by each corresponding receiver segmentpair, thereby providing an independent frequency spectrum of acousticreflection coefficients for each angular arc of the borehole.
 11. Themethod of claim 7 and further including the step of correcting saidsignals representative of discrete frequency components of therepresentative signals for any amplitude distortion introduced by thefrequency response of the acoustic receivers by amplifying each signalrepresentative of a discrete frequency component by an amount inverselyproportional to the sensitivity of the acoustic receiver in thefrequency region said signal represents.
 12. A method of acousticallylogging a cased borehole with a longitudinally extending exploringdevice to provide a plurality of frequency spectra of acousticreflection coefficients comprising the steps of: transmitting acomponent of acoustic energy radially outwardly frOm the exploringdevice into the cement and surrounding formations; receiving a reflectedportion of the transmitted acoustic energy at two different positionswithin the borehole which are at two positions having different radiirelative to the central axis of the exploring device and generatingfirst and second electrical signals representative thereof; separatingsaid first and second signals into a plurality of corresponding pairs offrequency component signals; and determining from said plurality ofcorresponding pairs of frequency component signals, the plurality ofacoustic reflection coefficients corresponding to each of the pairs offrequency component signals, and generating signals representativethereof.
 13. The method of claim 12 and further including the step ofrecording said plurality signals representative of the acousticreflection coefficient corresponding to each of the frequency componentsignals.
 14. The method of claim 12 wherein the steps are performedperiodically as a well tool is moved through the borehole and thesignals are recorded as a function of borehole depth.
 15. Apparatus foracoustically logging a cased well bore to provide an acoustic reflectioncoefficient spectrum comprising: an acoustic well logging tool having atleast one acoustic transmitter and at least two acoustic receivers ofdifferent radii; means for periodically firing said transmitter topropagate a component of acoustic energy radially outwardly from theborehole into the cement and surrounding formations; means forperiodically activating said acoustic receivers in response to saidtransmitter firings to receive a reflected portion of the transmittedenergy at said different radii and to generate a plurality of electricalsignals representative thereof; means for separating said plurality ofrepresentative electrical signals into a plurality of signalsrepresentative of discrete frequency components of said signals, saidfrequency components being in a one-to-one correspondence with eachother; means for generating electrical signals representative of therelative phases of corresponding pairs of said frequency componentsignals; means for generating electrical signals representative of therelative amplitudes of corresponding pairs of said frequency componentsignals; and means for combining said relative phase signals and saidrelative amplitude signals to generate signals representative of theacoustic reflection coefficient in the frequency region of each of saiddiscrete frequency components.
 16. The apparatus of claim 15 whereinsaid means for separating said plurality of representative electricalsignals into signals representative of discrete frequency componentsincludes a plurality of relatively narrow bandpass filters havingdifferent center frequencies disposed over a selected portion of theacoustic spectrum.
 17. The apparatus of claim 15 wherein said means forgenerating signals representative of the relative phases ofcorresponding pairs of said frequency component signals includes phasecomparator means for comparing the phases of said signals and means forgenerating an output signal representative of the difference of thephases of said signals.
 18. The apparatus of claim 15 wherein said meansfor generating electrical signals representative of the relativeamplitudes of corresponding pairs of said frequency component signalsincludes means for comparing the ratio of the amplitudes of said signalsand means for generating an output signal representative of saidamplitude ratio.
 19. The apparatus of claim 15 wherein said means forcombining said relative phase and said relative amplitude signalsincludes means for combining a signal representative of the differenceof the phases of said signals and the ratio of said amplitudes, togetherwith means for generating signals representative of the acousticreflection coefficient derived from the phase difference and amplituderatio siGnals.
 20. Acoustic apparatus for logging a well bore to provideacoustic reflection coefficient logs of the well bore comprising:acoustic transmitting transducer means; first piezoelectric acousticreceiver means of a hollow cylindrical shape having a radius r1; secondpiezoelectric acoustic receiver means of a hollow cylindrical shapehaving a radius r2 different from r1 and electrically distinct from saidfirst acoustic receiver; and means adapted for repetitively energizingsaid transmitting transducer means and producing separate electricalsignals from said first and second receiver means for use in producingacoustic reflection coefficient logs.
 21. The apparatus of claim 20wherein both of said acoustic receiver means are mounted on the exteriorsurface of plural cylinders of acoustic absorbent material.
 22. Theapparatus of claim 21 wherein both of said acoustic receiver means aredivided into n, where n is an integer number 2 or greater, correspondingangular segments, whereby each of the n corresponding receiver segmentpairs receive reflected acoustic energy from the angular portion of theborehole in front of the receiver segment pairs.
 23. The apparatus ofclaim 20 wherein said first and second receiver means are disposed alongthe longitudinal axis of the transducer array on either side of thetransmitting transducer in a zero spacing arrangement.
 24. Apparatus foracoustically logging a cased well borehole to determine theeffectiveness of cement disposed in the annulus between the casing andthe formation comprising: means for transmitting a component of acousticenergy radially outwardly from the borehole into the cement andsurrounding formations; means for receiving a reflected portion of thetransmitted acoustic energy at two different radial positions in theborehole and generating electrical signals representative thereof; meansfor performing a Fourier analysis of said first and second electricalsignals to separate said signals into a plurality of corresponding pairsof frequency component signals; and means responsive to said pluralityof corresponding pairs of frequency component signals for determiningthe plurality of acoustic reflection coefficients corresponding to saidfrequency component signals and generating signals representativethereof.
 25. The apparatus of claim 24 and further including means forrecording as a function of borehole depth said signals representative ofthe acoustic reflection coefficients corresponding to said pairs offrequency component signals.